Functional View of Geometry Processing: Operator-based Techniques for Shape Analysis
Résumé
In this document, we present a set of tools and algorithms for analyzing and processing shapes and their relations. Our main observation is that a very productive way of looking at many concepts in geometry processing, both in theory and in practice, is by representing them as linear operators acting on real-valued functions defined on the shapes. Although this point of view has been common in some areas of mathematics, such as dynamical systems, representation theory or parts of differential geometry, it has only recently been adopted in digital geometry processing, where it has led to novel insights and efficient algorithms for a wide variety of problems including shape matching, tangent vector field analysis and shape comparison to name a few. In this document, we will give an overview of these and related techniques and demonstrate, in particular, how the operator point of view can be helpful in a wide variety of practical settings, both by providing a common language in which many operations can be expressed and by enabling the use of classical linear-algebraic tools in novel, and sometimes unexpected scenarios.
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